Superalgebra

A $ \mathbf Z / 2 $- graded algebra over a field $ k $( see Graded algebra), i.e. a super-space $ A $ over $ k $ endowed with an even linear mapping $ A \otimes A \rightarrow A $. A superalgebra is said to be commutative (graded-commutative or supercommutative) if

$$ ab = (-1) ^ {p(a)p(b)} ba,\ \ a, b \in A; $$

here, $ p $ is a parity, i.e. a $ \mathbf Z /2 $- grading.

The definition of a superalgebra can be generalized to include the case where the domain of scalars is an arbitrary commutative associative superalgebra $ C $.

Examples of associative superalgebras over $ C $ are: the algebra $ M _ {m\mid n } (C) $ of matrices of the form

$$ \left ( \begin{array}{ll} X & Y \\ Z & T \\ \end{array} \right ) , $$

where $ X \in M _ {m} (C) $, $ T \in M _ {n} (C) $, endowed with the natural $ \mathbf Z / 2 $- grading (cf. Super-space); the tensor algebra $ T(M) $ of a $ \mathbf Z / 2 $- graded module $ M $ over $ C $; the symmetric algebra $ S(M) = T(M)/I $ of a module $ M $, where $ I $ is the ideal generated by the elements of the form

$$ x \otimes y-(-1) ^ {p(x)p(y)} y \otimes x; $$

and the exterior algebra $ \Lambda (M) = S( \Pi (M)) $ of a module $ M $( the latter two superalgebras are commutative).

A superalgebra $ \mathfrak G $ with a multiplication $ [ \cdot , \cdot ] $ is called a Lie superalgebra if for all $ x, y, z \in \mathfrak G $,

$$ [x, y] = (-1) ^ {p(x)p(y)+1} [y, x], $$

$$ [x, [y, z]] = [[x, y], z] + (-1) ^ {p(x)p(y)} [y, [x, z]] $$

(and $ [x, [x, x]]=0 $ if $ p(x)= \overline{1}\; $ and $ \mathop{\rm char} k = 3 $). In particular, there are no Lie superalgebras in characteristic $ 2 $, only $ \mathbf Z /2 $- graded Lie algebras.

Examples. Any associative superalgebra endowed with commutation (the supercommutator difference)

$$ [x, y] = xy - (-1) ^ {p(x)p(y)+1} yx $$

as the bracket operation; the algebra $ \mathop{\rm Der} A $ of derivations of an arbitrary superalgebra $ A $( i.e. of linear transformations $ \delta : A \rightarrow A $ for which $ \delta (ab) = ( \delta a)b + (-1) ^ {p( \delta )p(a) } a( \delta b) $) with the operation of commutation. For any Lie superalgebra $ \mathfrak G $ there is an associative universal enveloping superalgebra, and the straightforward generalization of the Birkhoff–Witt theorem holds.

The classification of finite-dimensional simple Lie superalgebras over the field $ \mathbf C $ is known (see [2], [3]). They are divided into Lie superalgebras of classical type (characterized by the fact that the Lie algebra $ \mathfrak G _ {0} $ is reductive) and Lie superalgebras of Cartan type. The Lie superalgebras of classical type are exhausted by the following series of matrix algebras:

$$ \mathop{\rm sl} (m, n) = \left \{ {\left ( \begin{array}{cc} X & Y \\ Z & T \\ \end{array} \right ) \in M _ {m\mid n} ( \mathbf C ) } : { \mathop{\rm Tr} X = \mathop{\rm Tr} T } \right \} \ \ (m \neq n); $$

$$ \mathop{\rm psl} (n, n) = { \mathop{\rm sl} (n, n) } / {\{ cE: c \in \mathbf C \} } ; $$

$$ \mathop{\rm osp} (m, 2n) = $$

$$ = \ \{ \alpha \in M _ {m\mid 2n} ( \mathbf C ): \beta ( \alpha (x), y) + (-1) ^ {p( \alpha ) p(x) } \beta (x, \alpha (y)) = 0 \} $$

for an even symmetric non-degenerate bilinear form $ \beta $;

$$ \mathop{\rm pe} (n) = \{ {\alpha \in M _ {n\mid n} ( \mathbf C ) } : {\beta ( \alpha (x), y) + \beta (x, (-1) ^ {p( \alpha ) p(x) } \alpha (y)) = 0 } \} $$

for an odd symmetric non-degenerate bilinear form $ \beta $;

$$ \mathop{\rm spe} (n) = \mathop{\rm pe} (n) \cap \mathop{\rm sl} (n, n), $$

$$ \mathop{\rm q} (n) = \left \{ \left ( \begin{array}{cc} X & Y \\ Y & X \\ \end{array} \right ) \in M _ {n\mid n} ( \mathbf C ) : \mathop{\rm Tr} Y=0 \right \} = $$

$$ = \ \left \{ \alpha \in M _ {n\mid n} ( \mathbf C ) : \left [ \alpha ,\ \left ( \begin{array}{rl} 0 &1 _ {n} \\ -1 _ {n} & 0 \\ \end{array} \right ) \right ] =0 \right \} , $$

$$ \mathop{\rm sq} (n) = \{ \alpha \in \mathop{\rm q} (n) : q \mathop{\rm Tr} \alpha = 0 \} , $$

where

$$ q \mathop{\rm Tr} \left ( \begin{array}{cc} A & B \\ B & A \\ \end{array} \right ) = \mathop{\rm Tr} B \left ( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right ) , $$

$$ \mathop{\rm psq} (n) = \mathop{\rm sq} (n) / {\{ cE: c \in \mathbf C \} } , $$

and certain exceptional algebras (of dimensions $ 17 $, $ 31 $ and $ 40 $). The superalgebras of Cartan type are the algebra $ \mathop{\rm Der} \Lambda ( \mathbf C ^ {n} ) $ and its supersubalgebras, analogues to the simple Lie graded algebras $ W _ {n} $, $ S _ {n} $, $ H _ {n} $( cf. Lie algebra, graded).

The classification of real structures of simple Lie superalgebras and a description of semi-simple Lie superalgebras in terms of simple ones are also known.

The theory of linear representations of Lie superalgebras is essentially more complex than for Lie algebras in that representations of simple Lie superalgebras, as a rule, are not completely reducible, while irreducible representations of solvable Lie superalgebras need not be one-dimensional. A classification exists of the irreducible representations of simple finite-dimensional Lie superalgebras over $ \mathbf C $ in terms of the highest weights (see , [2]), and an explicit description is known of the finite-dimensional representations, as well as for the character formula for certain series of these algebras .

References

[1a]	D.A. Leites, "Lie superalgebras" Josmar , 30 (1984)
[1b]	D.A. Leites (ed.) , Seminar on supermanifolds , Kluwer (1990)
[2]	V.G. Kac, "Lie superalgebras" Adv. Math. , 26 (1977) pp. 8–96
[3]	M. Scheunert, "The theory of Lie superalgebras. An introduction" , Springer (1979)

Comments

A classification of the simple finite-dimensional Lie superalgebras over $ \mathbf C $ was obtained by V.G. Kac in 1975 (see [2]).

A classification of the irreducible finite-dimensional representations of solvable Lie superalgebras may be found in [2].

The irreducible finite-dimensional representations of a simple Lie superalgebra are divided into two classes: typical and atypical (exceptional). Characters of typical representations were computed in [a1]. Characters of atypical representations are not known, not even in the case of $ \mathop{\rm sl} (m, n) $.

References

[a1]	V.G. Kac, "Representations of classical Lie superalgebras" K. Bleuler (ed.) et al. (ed.) , Differential Geometrical Methods in Mathematical Physics II , Lect. notes in math. , 676 , Springer (1978) pp. 597–626
[a2]	F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds)
[a3]	F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian)